As every problem \(A\)  is a set of binary strings (words) \(x\) , and as every reduction from \(A\)  to \(B\)  is a function \(f\)  such that \(x\in A\)  iff \(f(x)\in B\) , we have that the reductions described above are standard in the sense that only a few bits are required from \(x\)  in order to determine the identity of the \(j\) 'th bit of \(f(x)\) . For example, in the reduction from \(3SAT \)  to \(CLIQUE \) , the output is a binary string that represents an adjacency matrix of a graph of size polynomial to the size of the input (and this size can be found in logarithmic time, see [1]}). The content of every cell \((i,j)\)  of that matrix can be determined by whether two satisfying assignments to 3SAT, one for a certain clause \(c\)  and one for a certain clause \(c^{\prime }\)  in the input share the same variable (see [2]} for more details). In this reduction, the output formula is standard, in the sense that the location of the clauses \(c,c^{\prime }\)  in the input string is easily found. As such, determining a certain output bit requires only a small (logarithmic) access to the input string, followed by a small (also logarithmic) comparison of the obtained input process.
Similarly, the Cook-Levin reduction from any NP problem to \(SAT\)  is standard in that sense as well, and therefore the output as the SAT formula can be obtained from logarithmic input and comparison. See [3]} for more details.
